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Metamath Proof Explorer

Theorem gsummptfsres

Description: Extend a finitely supported group sum by padding outside with zeroes. (Contributed by Thierry Arnoux, 15-Feb-2026)

Ref Expression
Hypotheses gsummptfsres.1 ⊢ 𝐵 = ( Base ‘ 𝐺 )
gsummptfsres.2 ⊢ 0 = ( 0g ‘ 𝐺 )
gsummptfsres.3 ⊢ ( 𝜑 → 𝐺 ∈ CMnd )
gsummptfsres.4 ⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
gsummptfsres.5 ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐴 ∖ 𝑆 ) ) → 𝑌 = 0 )
gsummptfsres.6 ⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝑌 ) finSupp 0 )
gsummptfsres.7 ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝑌 ∈ 𝐵 )
gsummptfsres.8 ⊢ ( 𝜑 → 𝑆 ⊆ 𝐴 )
Assertion gsummptfsres ( 𝜑 → ( 𝐺 Σg ( 𝑥 ∈ 𝐴 ↦ 𝑌 ) ) = ( 𝐺 Σg ( 𝑥 ∈ 𝑆 ↦ 𝑌 ) ) )

Proof

Step Hyp Ref Expression
1 gsummptfsres.1 ⊢ 𝐵 = ( Base ‘ 𝐺 )
2 gsummptfsres.2 ⊢ 0 = ( 0g ‘ 𝐺 )
3 gsummptfsres.3 ⊢ ( 𝜑 → 𝐺 ∈ CMnd )
4 gsummptfsres.4 ⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
5 gsummptfsres.5 ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐴 ∖ 𝑆 ) ) → 𝑌 = 0 )
6 gsummptfsres.6 ⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝑌 ) finSupp 0 )
7 gsummptfsres.7 ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝑌 ∈ 𝐵 )
8 gsummptfsres.8 ⊢ ( 𝜑 → 𝑆 ⊆ 𝐴 )
9 7 fmpttd ⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝑌 ) : 𝐴 ⟶ 𝐵 )
10 5 4 suppss2 ⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐴 ↦ 𝑌 ) supp 0 ) ⊆ 𝑆 )
11 1 2 3 4 9 10 6 gsumres ⊢ ( 𝜑 → ( 𝐺 Σg ( ( 𝑥 ∈ 𝐴 ↦ 𝑌 ) ↾ 𝑆 ) ) = ( 𝐺 Σg ( 𝑥 ∈ 𝐴 ↦ 𝑌 ) ) )
12 8 resmptd ⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐴 ↦ 𝑌 ) ↾ 𝑆 ) = ( 𝑥 ∈ 𝑆 ↦ 𝑌 ) )
13 12 oveq2d ⊢ ( 𝜑 → ( 𝐺 Σg ( ( 𝑥 ∈ 𝐴 ↦ 𝑌 ) ↾ 𝑆 ) ) = ( 𝐺 Σg ( 𝑥 ∈ 𝑆 ↦ 𝑌 ) ) )
14 11 13 eqtr3d ⊢ ( 𝜑 → ( 𝐺 Σg ( 𝑥 ∈ 𝐴 ↦ 𝑌 ) ) = ( 𝐺 Σg ( 𝑥 ∈ 𝑆 ↦ 𝑌 ) ) )